Angelo Vistoli in the notes Notes on Grothendieck topologies, fibered categories and descent theory starts the section of category theory with the following note:

We will not distinguish between small and large categories. More generally, we will ignore any set-theoretic difficulties. These can be overcome with standard arguments using universes.

Question : Which of the notions introduced in Angelo Vistoli's notes assumes that the category is small? In particular their application to Algebraic/differentiable/topological stacks?

For example, Behrang Noohi puts the following extra condition in his notes on topological stacks:

Throughout the paper, all topological spaces are assumed to be compactly generated.

This could be because, the category $\text{Top}$ of all topological spaces is not a small category.

Are there any places one has to be careful to not allow large categories?

Angelo Vistoli in the notes Notes on Grothendieck topologies, fibered categories and descent theory starts the section of category theory with the following note:

We will not distinguish between small and large categories. More generally, we will ignore any set-theoretic difficulties. These can be overcome with standard arguments using universes.

Question : Which of the notions introduced in Angelo Vistoli's notes assumes that the category is small? In particular their application to Algebraic/differentiable/topological stacks?

For example, Behrang Noohi puts the following extra condition in his notes on topological stacks:

Throughout the paper, all topological spaces are assumed to be compactly generated.

This could be because, the category $\text{Top}$ of all topological spaces is not a small category.

Are there any places one has to be careful to not allow large categories?

Some references to support this question :

1. nlab says "In technical terms, a site is a small category equipped with a coverage or Grothendieck topology". It also says (Remark $2.3$ at same page) "Often a site is required to be a small category. But also large sites play a role."
2. David Metzler in Topological and Smooth Stacks defines (page $2$) a site as a small category equipped with Grothendieck topology. It further says "We will want to discuss, for example, “the category of stacks on the category of all topological spaces,” but strictly speaking this does not exist, since the category of topological spaces does not have a set of objects, but rather a proper class. To avoid this problem we will consider throughout some fixed category $\mathbb{T}$ of topological spaces which has a set of objects, or at least, is equivalent to such a category".

So, it "looks like", even though one can define a site over a large category, and then a stack over a site (which was defined on a large category), one often restricts (for computational purposes or personal interests) to a small categories and stacks on them. Is this what it is or am I misunderstanding something here?

Angelo Vistoli in the notes Notes on Grothendieck topologies, fibered categories and descent theory starts the section of category theory with the following note:

We will not distinguish between small and large categories. More generally, we will ignore any set-theoretic difficulties. These can be overcome with standard arguments using universes.

Question : Which of the notions introduced in Angelo Vistoli's notes assumes that the category is small? In particular their application to Algebraic/differentiable/topological stacks?

For example, Behrang Noohi puts the following extra condition in his notes on topological stacks:

Throughout the paper, all topological spaces are assumed to be compactly generated.

This could be because, the category $\text{Top}$ of all topological spaces is not a small category.

Are there any places one has to be careful to not allow large categories?

Angelo Vistoli in the notes Notes on Grothendieck topologies, fibered categories and descent theory starts the section of category theory with the following note:

We will not distinguish between small and large categories. More generally, we will ignore any set-theoretic difficulties. These can be overcome with standard arguments using universes.

Question : Which of the notions introduced in Angelo Vistoli's notes assumes that the category is small? In particular their application to Algebraic/differentiable/topological stacks?

For example, Behrang Noohi puts the following extra condition in his notes on topological stacks:

Throughout the paper, all topological spaces are assumed to be compactly generated.

This could be because, the category $\text{Top}$ of all topological spaces is not a small category.

Are there any places one has to be careful to not allow large categories?